Would a Senate acquittal moot some of the impeachment-related subpoena cases?
In theory, at least, the House could continue to investigate possible impeachment proceedings indefinitely, even after an acquittal
The House of Representatives has issued several subpoenas related to its impeachment inquiry. The Supreme Court may resolve the validity of some of these subpoenas in the near future. Other subpoenas, which are not on expedited tracks, would not be resolved until after the 2020 election.
What happens to the latter category of subpoenas if the President is impeached, and subsequently acquitted in the Senate of all charges? Could the President move to dismiss all pending cases as moot?
Perhaps the House would counter that it needs the information to consider another impeachment inquiry. After all, the Double Jeopardy Clause of the Fifth Amendment does not control the impeachment process. (Ditto for the Sixth Amendment.) In theory, the House could enter into a permanent impeachment inquiry–keep searching for new evidence until it can make the case to convict the President.
If the cases do become moot, then many of these cases may never be resolved by the courts.
I highly recommend Charlie Savage's article in the Times. He details how the Trump administration can win, merely by pushing the litigation deadlines beyond the scope of a Senate trial.